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          <h1 class="post-title" itemprop="name headline">GIS算法基础（一） 计算几何基础（上）

            
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              <time title="创建时间：2018-10-08 15:35:52" itemprop="dateCreated datePublished" datetime="2018-10-08T15:35:52+08:00">2018-10-08</time>
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        <p>最近在学习GIS算法，在学习过程中，想把一些经典的算法或者思想记录下来，分享给大家</p><p>计算几何基础本来是计算机图形学的内容，但是GIS在图像处理中是离不开计算机处理的，所以GIS算法基础第一个应该是计算几何基础。</p><p>如何把空间实体的点线面以及他们之间的关系（例如，相交，包含，邻接，叠置等等）用计算机语言表达出来，应该是这里解决的第一个问题。前人已经有过这样这样的解决模型—DE-9IM模型（<strong>Dimensionally Extended nine-Intersection Model</strong>）</p><a id="more"></a>


<p>DE9IM模型是一种拓扑模型，他用于描述多边形之间的关系。现有两个多边形，如果他们相交，那他们的边界，内部，外部的交集情况是怎么样的？如果相离呢？情况又是怎样的？DE9IM实际上就是一个3X3的矩阵，他存储着两个多边形，边界，内部，外部，的交集情况。</p>
<p>假如交集的结果用维数dim ={-1,0,1,2} 表示，-1代表无交集，0代表交集为一个点，1代表线，2代表面。</p>
<p><img src="https://zhong-blog.oss-cn-shenzhen.aliyuncs.com/blog/20190907202739.png!blog" alt="20190907202739"></p>
<p>上面这个图能很好的说明这个问题。 a,b代表两个面，他们边界，内部，外部，相交的结果就是 红色的部分</p>
<p>例如a的内部和b的内部  交集应该是一个面，所以是2；a的边界和b的边界，交集应该是两个点，所以是0；</p>
<p>总结一下：<strong>dim = -1 无交集;</strong></p>
<p>​                  <strong>dim = 0 交集中不包含线和面，但至少包含一个点</strong></p>
<p>​                  <strong>dim = 1  交集中不包含面，但至少包含一个面</strong></p>
<p>​                  <strong>dim = 2 交集至少包含一个面</strong></p>
<p> 现在多边形之间的问题的模型解决了，那么如何用计算机语言表达呢？</p>
<p>上面也说了，这个交集的结果可以用一个3x3的矩阵表达。例如上边的a,b交集结果 表达为：</p>
<p><strong><em>2 1 2</em></strong></p>
<p><strong><em>1 0 1</em></strong></p>
<p><strong><em>2 1 2</em></strong></p>
<p>于是，空间关系的描述可以归纳为：两个几何体，以表示两个几何体的DE-9IM结果的矩阵输入，然后把这个矩阵转为字符串，例如上面这个就变为”212101212”了</p>
<p>如果我们说两个多边形相交，那就是他们的结果要有 0,1,2 也就是交集中游点或线或面</p>
<p>在计算机中就是 (0,1,2) -&gt;true</p>
<p>​                           (-1) -&gt;false</p>
<p>​                            (-1,0,1,2)-&gt;*</p>
<p>那么上面那个字符串就可以表示成”TTTTTTTTT”</p>
<p>经过前人的整理，空间关系的判断可以通过特定的一些模式进行判断</p>
<table>
<thead>
<tr>
<th>谓词</th>
<th>返回值</th>
<th>描述</th>
</tr>
</thead>
<tbody><tr>
<td>相等（Equals)</td>
<td><code>T*F**FFF*</code></td>
<td>边界上的点和内部的点全部重合。属于相交的一种</td>
</tr>
<tr>
<td>相离（Disjoint)</td>
<td><code>FF*FF****</code></td>
<td>不相交，与相交相反</td>
</tr>
<tr>
<td>相接（Touches)</td>
<td><code>FT*******\F**T*****\F***T****</code></td>
<td>只有边界上有共同点，内部没有。属于相交的一种</td>
</tr>
<tr>
<td>覆盖（Covers)</td>
<td><code>T*****FF*\*T****FF*\***T**FF*\****T*FF*</code></td>
<td>b上的每个点都在a上（边界和内部），且所有点都不在a外部。属于相交的一种</td>
</tr>
<tr>
<td>叠置（Overlaps)</td>
<td><code>T*T***T**\1*T***T**</code></td>
<td>a和b相交，且具有一部分共同点，但不是全部内部点。属于相交的一种</td>
</tr>
</tbody></table>
<p>所以，只要我们得到了两个多边形的DE9IM矩阵结果，我们就可以判断他们之间的关系了。</p>
<p>那怎样获得两个多边形的交集结果呢？这关系到点，线，面-点，线，面 之间的判断，这些后面在更</p>

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